Two numbers are in the ratio 5: 7. On diminishing each of them by 40, they become in the ratio 17 : 27. The difference of the numbers is :

Options

A. 18
B. 52
C. 137
D. 50

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \). According to the problem, the ratio of these two numbers is given as:
\[ \frac{x}{y} = \frac{5}{7} \]

### Step 2: Express \( y \) in terms of \( x \)
From the ratio, we can express \( y \) in terms of \( x \):
\[ 7x = 5y \implies y = \frac{7}{5}x \]

### Step 3: Set Up the Second Ratio
When each number is diminished by 40, the new ratio becomes:
\[ \frac{x - 40}{y - 40} = \frac{17}{27} \]

### Step 4: Cross Multiply to Form an Equation
Cross multiplying gives us:
\[ 27(x - 40) = 17(y - 40) \]
Expanding both sides:
\[ 27x - 1080 = 17y - 680 \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ 27x - 17y = 400 \quad \text{(Equation 1)} \]

### Step 6: Substitute \( y \) in Equation 1
Now substitute \( y = \frac{7}{5}x \) into Equation 1:
\[ 27x - 17\left(\frac{7}{5}x\right) = 400 \]
This simplifies to:
\[ 27x - \frac{119}{5}x = 400 \]

### Step 7: Clear the Fraction
To eliminate the fraction, multiply the entire equation by 5:
\[ 135x - 119x = 2000 \]
This simplifies to:
\[ 16x = 2000 \]

### Step 8: Solve for \( x \)
Now, divide both sides by 16:
\[ x = \frac{2000}{16} = 125 \]

### Step 9: Find \( y \)
Substituting \( x = 125 \) back into the equation for \( y \):
\[ y = \frac{7}{5} \times 125 = 175 \]

### Step 10: Calculate the Difference
Now, we can find the difference between the two numbers:
\[ y - x = 175 - 125 = 50 \]

### Final Answer
The difference of the numbers is \( \boxed{50} \).

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